In what ways is the term "topology" applied to Artificial Intelligence?
In addition to the ways the term topology is itself used generically to describe the "shape" of various aspects of Machine Learning, the term appears in the field Topological Data Analysis:


  In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.


Some examples of its use in ML:


The ring of algebraic functions on persistence bar codes, Homology, Homotopy and Applications
Persistence Images: A Stable Vector Representation of Persistent Homology
Topological data analysis and machine learning theory
Topological Kernel Learning: Discrete Morse Theory is used to connect kernel machine learning with topological data analysis.